Conformal approach to cylindrical DLA

TL;DR

This paper extends the conformal mapping approach to cylindrical DLA, enabling analysis of the relationship between radial and cylindrical geometries through a novel complex function, revealing self-affine features and particle size dynamics.

Contribution

It introduces a complex function that connects radial and cylindrical DLA, allowing for new quantitative studies of their relationship and geometric properties.

Findings

Cylindrical DLA exhibits self-affine features similar to radial DLA.

A complex transformation links radial and cylindrical geometries.

Particle size variation explains the relationship between the two geometries.

Abstract

We extend the conformal mapping approach elaborated for the radial Diffusion Limited Aggregation model (DLA) to the cylindrical geometry. We introduce in particular a complex function which allows to grow a cylindrical cluster using as intermediate step a radial aggregate. The grown aggregate exhibits the same self-affine features of the original cylindrical DLA. The specific choice of the transformation allows us to study the relationship between the radial and the cylindrical geometry. In particular the cylindrical aggregate can be seen as a radial aggregate with particles of size increasing with the radius. On the other hand the radial aggregate can be seen as a cylindrical aggregate with particles of size decreasing with the height. This framework, which shifts the point of view from the geometry to the size of the particles, can open the way to more quantitative studies on the relationship between radial and cylindrical DLA.